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Matters for IB considerationOur research into use of technology in 16-19 academic stream mathematics leads us to twosets of matters for IB consideration in their coming curriculum review of DP mathematics:tools and resources; and pedagogic practices and professional development.Tools and resourcesWhilst the IB was innovative in the past in embracing graphic calculator use, DPmathematics does, perhaps, place too much emphasis on this single tool. Coupled with this,tool use should be seen in the context of the resources available in classrooms. A challengefor the IB is to consider a wide range of tools and resources available for DP mathematics:traditional mathematical tools; techno-mathematical tools with capabilities for algebra,geometry and calculus; traditional resources such as textbooks; and digital/internet-basedresources for teaching and learning 2.There are many sub-challenges within this challenge. The use of tools with capabilities foralgebra, geometry and calculus would require serious consideration of the place oftechniques and concepts in DP mathematics and it may be useful to consider parallel byhand/by-technology DP courses so that teachers may come on board when they areconfident 3. Linking graphic calculator use to the full range of classroom resources wouldinclude reviewing the tasks teachers offer to students and this may lead the IB to reconsiderwhether textbooks should be included in IB documentation. Consideration of internet-basedresources may lead to on-line communities of teachers and students and the challenge offacilitating these communities.Whilst IB staff who are not classroom-based have an obvious role in taking these challengesforward, it is important that teachers are a full part of finding solutions to these challenges,which leads us on to the next set of matters for IB consideration.Pedagogic practices and professional developmentThe quality of IB teacher support material and workshops appears very high but the IBshould consider whether this is sufficient for teacher development for the integration oftechnology into DP mathematics. The success of designs for such integration requires accessto technology, the development of technological knowledge and assisting teachers todevelop modes of organising classrooms for student learning. Our review of research pointsto the benefits of teachers’ collaborative work in this endeavour.A model of professional development that has had some success is to structure it around asupportive community of inquiry on everyday classroom practice where all participants areco-learners and knowledge, classroom practices and resources are developed, shared andevaluated by the group. Given the geographical distance between IB schools on-linecommunities can be set up to facilitate such communities of inquiry.23See Scope of Work 1, section 1(b) for details.See Scope of Work 1, section 3(a) for details.7

Scope of Work 1: Literature ReviewMethodIn this review we made a decision to concentrate on papers written from the years 2000 to2014 (exceptions were made for key papers) from journals rated as A* and A (and somerated as B) in a recent European rating of mathematics education journals (seehttp://www.mathematik.uni-dortmund.de/ erme/). The journals are: Educational Studies in MathematicsJournal of Mathematical BehaviorJournal for Research in Mathematics EducationMathematical Thinking and LearningMathematics Education Research JournalRecherches en Didactique des MathématiquesResearch in Mathematics EducationZDM: The International Journal on Mathematics EducationWe considered every paper in these journals for 2000 – 2014 and included all papers in ourreview that deal with technology which impacts in some way on 16-19 mathematics. Somepapers were included which were not specifically focused on but were felt to impact on 1619 mathematics with technology. Papers focussing exclusively on elementary mathematicswith technology were excluded from our review.In addition we looked at a number of important books as well as adding relevant papers thatwere known to us, or written by us. These included:Two International Handbooks of Mathematics Education:Bishop, A.J., Clements, M.A., Keitel, C., Kilpatrick, J. & Leung, F.K.S. (Eds.) (2003).Second international handbook of mathematics education. Dordrecht: KluwerAcademic Publishers.Bishop, A.J., Clements, M.A., Keitel, C., Kilpatrick, J. & Leung, F.K.S. (Eds.) (2013).Third international handbook of mathematics education. New York: Springer.Two books taking a general look at ICT integration in mathematics education:Guin, D., Ruthven, K. &Trouche, L. (Eds.) (2005), The didactical challenge of symboliccalculators: turning a computational device into a mathematical instrument.Gueudet, G., Pepin, B. &Trouche, L. (Eds.) (2012), From text to ‘lived’ resources:Mathematics curriculum materials and teacher development. New York:Springer.The 17th ICMI Study:Hoyles, C. &Lagrange, J.-B. (Eds.). (2010). Mathematical education and digitaltechnologies: Rethinking the terrain. New York: Springer.A book on teacher use of ICT:Clark-Wilson, A. Sinclair, N. & Robutti, O. (Eds.) (2014). The mathematics teacher inthe digital era. Dordrecht: Springer.8

The references are listed in the appendix for SoW 1 as is a bibliography that includes papersnot included in this review but which may be useful.The structure of the literature review is as follows:1. Teacher Issues and Technology Usea)b)c)d)e)Some Theoretical ConsiderationsApplications of the Theory to Classroom PracticeAffordances and Constraints in Teaching with TechnologyProfessional Development of TeachersEmerging Practice: Challenges and Implications for Curriculum Change2. Student Issues and Technology Usea) Student Learning Outcomesb) Examples of Technology Use and Specific Curriculum Areasi. Tasksii. Geometryiii. Statisticsc) Student and Teacher Interactionsd) Classroom Organisation for Student Learninge) Emerging Practice, Challenges and Implications for Curriculum Change3. Technology and Assessmenta) Formative and Summative Assessmentb) Emerging Practice, Challenges and Implications for Curriculum Change4. Some Implications and Predictions Related to Curriculum ChangeFindings1. Teacher Issues and Technology Usea) Some Theoretical ConsiderationsThere have been a number of theoretical approaches applied to issues surroundingtechnology use in the secondary mathematics classroom and the last decade has seenefforts aimed at using these frameworks to provide different, complementary views of thesame subject. For example, Lagrange et al. (2003), through a review of 800 articles,distinguished seven dimensions related to ICT use in mathematics classrooms: integration,epistemological-semiotic, cognitive, institutional, instrumental, situational and the teacherdimension. While this last dimension appears to have been quite neglected during the years1990-2000, it has increasingly been taken into account in the last decade, including a recent9

book (Clark-Wilson, Robutti & Sinclair, 2014) devoted to the mathematics teacher in thedigital era. In addition, Drijvers et al. (2010), have suggested the need for an integrativeframework in which considerations of didactical functionalities play an essential role,combining the three major dimensions of tool features, educational goals and associatedpotential of the tool and modalities of use in a teaching/learning process.While it is clear that the role of the teacher is a key to the successful use of digitaltechnology in the mathematics classroom, incorporating technology into teaching remains achallenge for many teachers and they need to be well prepared (Zbiek & Hollebrands, 2008).Hoyles et al. (2010) provide evidence for the critical role of teacher: as a facilitator whomaintained and supported the interaction, as a gatherer, making visible on a commonworkspace (the screen, for example) students’ production and as a mechanism fordiscussion; validating what did and did not make sense in terms of knowledge building.Some of the intrinsic factors contributing to the challenge facing teachers are theirorientations (Schoenfeld, 2011); their instrumental genesis (Artigue, 2002; Guin & Trouche,1999; Rabardel, 1995; Vérillon & Rabardel, 1995) and orchestrations (Trouche, 2005c);Drijvers & Trouche 2008); their perception of the nature of mathematical knowledge andhow it should be learned (Zbiek & Hollebrands, 2008); their mathematical contentknowledge; and their mathematical knowledge for teaching (MKT – Ball, Hill &Bass, 2005;Hill & Ball, 2004; Zbiek, Heid, Blume & Dick, 2007). The idea of MKT covers appropriatestructuring of content and relevant classroom discourse and activities to form didacticalsituations. Thus, it is not surprising that, while many mathematics teachers claim to supportthe use of technology in their teaching (Forgasz, 2006a; Thomas, 2006), the degree and typeof use in the classroom remains variable (Ruthven & Hennessy, 2002; Zbiek & Hollebrands,2008).Two complementary frameworks that have merit in analysing factors that may influenceteacher use of technology and provide an indication of teacher readiness forimplementation of technology use, are Pedagogical Technology Knowledge (PTK) (Hong &Thomas, 2006; Thomas & Hong, 2005b) and the Technological Pedagogical ContentKnowledge (TPACK) (Mishra & Koehler, 2006; Koehler & Mishra, 2009). A critical review ofthe notion of TPACK, including an analysis of its affordances and constraints can be found inGraham (2011). Both of these frameworks build on pedagogical content knowledge (PCK) ofShulman (1986), by adding technology knowledge. In this aspect, PTK incorporates theprinciples, conventions and techniques required to teach mathematics through thetechnology. This includes the need to be a proficient user of the technology, but moreimportantly, to understand the principles and techniques required to build and managedidactical situations incorporating it and enable mathematical learning through thetechnology. According to Koehler and Mishra (2009, p. 64), the TPACK definition oftechnology knowledge is close to that of Fluency of Information Technology (FITness), asproposed by the Committee of Information Technology Literacy of the National ResearchCouncil (see Snyder et al., 1999). This requires that teachers understand informationtechnology broadly enough to apply it productively at work and in their everyday lives, torecognise when information technology can assist or impede the achievement of a goal andcontinually to adapt to changes in information technology.10

b) Applications of the Theory to Classroom PracticeIn addition to technology knowledge, PTK includes teacher orientations and goals(Schoenfeld, 2011), especially beliefs about the value of technology and the nature oflearning mathematical knowledge as well as crucial affective aspects, such as teacherconfidence (Thomas & Palmer, 2014). Although in general teachers may agree there arepotential benefits of technology for students' mathematics learning, many remain unsure orunconvinced about whether its use helps students to explore mathematical concepts orunfamiliar problems (Forgasz, 2002; Thomas, 2006; Goos & Bennison, 2008). That teacherbeliefs can inhibit technology use, was demonstrated by Zuber and Anderson (2012) in aone-to-one laptop programme. They report that a prevalent belief limiting laptop use is thatstudents only learn mathematics authentically using pen and paper. Further, cautiousAdopters and Non Adopters of technology expressed beliefs that laptops exacerbateclassroom management problems, especially for lower-achieving students. The study byLagrange and Dedeoglu (2009), in the context of “ordinary” classes, reveals a high level ofteacher expectation with regard to technology, but a quite low degree of integration. Theyanalyse this phenomenon as the co-existence of two worlds, the world of teacherexpectation and the world of technology potentialities. To establish links between these twoworlds requires encouraging teachers’ collective work and reflective practices.One useful way to reflect on the alignment of the teacher preparedness factors with theiractual classroom practice is that of instrumental orchestrations (see Drijvers & Trouche,2008 for the theoretical framework of instrumentation and orchestration and someconcrete examples), or the ways in which the teacher manages the changing classroomenvironment when technology is present. A primary goal of the orchestrations is to engagestudents in activity producing techniques with both epistemic value, providing knowledge ofthe mathematical object under study and ‘productive potential’ or pragmatic value (Artigue,2002). The classroom orchestrations appear as a response to both the proliferation andincreasing complexity of technological resources. The notion of orchestration itself evolves,from an individual and static conception (orchestrations of a mathematical situation seenthrough didactical configurations and exploitation modes of these configurations – Trouche2002, 2005c, 2005d, 2007) with a social perspective (orchestrations seen as the result ofteachers collaborative work – Hoyles et al., 2010) to a dynamic view (including the didacticalperformance, teachers’ adaptation on the fly – Drijvers et al., 2010 and teacher adaptationover time – Drijvers, 2012). Some researchers (e.g., Drijvers, Doorman, Boon, Reed &Gravemeijer, 2010; Drijvers, Tacoma, Besamusca, Doorman & Boon, 2013) have taken theframework of instrumental orchestration as a point of departure for categorising observedteaching practices. This results in a kind of (non-exhaustive and maybe tool, topic andculture dependent) taxonomy, depicted in the Figure below. The Figure provides anoverview of whole-class and individual orchestrations identified in the study; the middlearrows simply show correspondence between whole-class and individual orchestrations.11

Other analyses of factors influencing teacher implementation of a technology-richmathematics curriculum and practices include that by Assude et al. (2010, p. 406), whopropose a typology comprising: the social, political economical and cultural level; themathematical and epistemological level; the school and institutional level; the classroomand didactical level. Further, Ruthven (2012) identifies five key structuring features ofclassroom practice and shows how they relate to the constitution of digital tools andmaterials as classroom resources: working environment (physical surroundings wherelessons take place), resource system (collection of mathematical tools and materials inclassroom use), activity format (generic templates for action and interaction which framethe contributions of teacher and students to particular types of lesson segment), curriculumscript (evolving teacher knowledge gained in the course of her own experience of learningand teaching a given topic) and time economy (managing the physical time available forclassroom activity to be converted into a didactical time, measured in terms of advance ofknowledge). Finally, a web metaphor has been used to describe the new way of teachingmathematics in the digital era (Leguay, 2011).There appears to be a need for teachers to evolve and adapt their teaching when usingtechnology. According to Doerr and Zangor (2000), the role, knowledge and beliefs of theteacher, along with the nature of the mathematical tasks, influenced the type of GDC useemerging, with five kinds identified: computational tool (evaluating numerical expressions,estimating and rounding), transformational tool (changing the nature of the task), datacollection and analysis tool (gathering data, controlling phenomena, finding patterns),visualizing tool (finding symbolic functions, displaying data, interpreting data, solvingequations) and checking tool (confirming conjectures, understanding multiple symbolicforms). Using their practitioner model, Ruthven and Hennessy (2002; 2003) provide aframework for synthetising teacher thinking about the contribution of GDC, with lateranalysis (Ruthven, Deaney & Hennessy, 2009) highlighting the crucial part played by teacherpre-structuring and shaping of technology-and-task-mediated student activity.An application of the concepts of Zone of Proximal Development (ZPD), Zone of FreeMovement (ZFM) and Zone of Promoted Action (ZPA) by Goos and others (Goos, 2005;Goos, Galbraith, Renshaw & Geiger, 2000, 2003) to describe teachers’ thinking and modesof working with technology, uses the metaphors of technology as master, servant, partnerand extension of self. In the last of these technology is “seamlessly incorporated into ateacher’s pedagogical and mathematical repertoire such as through the integration of a12

variety of technology resources into course planning and the everyday practices of themathematics classroom” (Goos, 2005, p. 40). Kendal et al. (2005) analyse the different waysin which three pioneering Australian teachers adapted their teaching to use CAS (computeralgebra system) and found it depended on their history and experiences. An example fromSinclair and Yurita (2008) focuses on changes to the instructional environment requiredwhen using a dynamic geometry system (DGS) with regard to the manner in which theteacher talks about geometric objects and uses visual artefacts and models of geometricreasoning.c) Affordances and Constraints in Teaching with TechnologySome papers address the affordances and constraints that either promote or inhibit theability of teachers to use technology. Heid et al. (2013) summarise a number of factors,including those in other research (see e.g., Forgasz, 2006a; Goos, 2005; Pierce & Ball, 2009;Thomas, 2006; Thomas et al., 2008; Thomas & Chinnappan, 2008) that influence teacheradoption and implementation of technology in mathematics teaching. They mention, forexample, “previous experience in using technology, time, opportunities to learn,professional development, access to technology, availability of classroom teachingmaterials, support from colleagues and school administration, pressures of curriculum andassessment requirements and technical supports” (Heid et al., 2013, p. 630). Lack ofaffordability of the technology typically used for mathematics teaching has equityimplications (Pierce & Ball, 2009), since the perceived advantages of this technology forsupporting student learning and examination use may not be equally available for allstudents. With regard to classroom resources, Kieran et al. (2012) analysed how affordancesand constraints are inherent in any given resource, noting “Resources are not neutral. Theyspeak to different teachers in different ways.” (p. 211). This influence on teacher use arisesfrom the designers’ intentions (both explicit and implicit).d) Professional Development of TeachersThere are clear implications arising from the literature with regard to teacher use oftechnology in mathematics. One major aspect is the nature of pre- and in-serviceprofessional development (PD) of teachers that will enhance their ability to incorporatetechnology into learning situations. It seems clear (Bennison & Goos, 2010) thatparticipation in professional development plays a crucial role in whether and howtechnology is used in mathematics classrooms. In this regard there are several relatedaspects that PD needs to consider: issues of access to technology in schools; the promotionand development of teacher technological knowledge (based on PTK or TPACK); andassisting teachers to broaden and deepen their engagement with instrumentalorchestrations. Research shows that teachers want PD that models planning and pedagogyso they can meaningfully integrate technology into their lessons in ways that help studentslearn mathematical concepts (Goos & Bennison, 2008).It may be that the goals of PD are best accomplished when it is structured around asupportive community of inquiry (Jaworski, 20001, 2003) where all participants are colearners and knowledge is developed and evaluated critically as a group. In the context of anational programme for providing digital resources to teachers, Trigueros and Lozano (2012,p. 261) acknowledge that “creating working groups inside and outside the schools constitutean essential element in professional development programs”. Being part of a group thatshares and reflects on their cont

key to the successful use of technology in the mathematics c lassroom. but incorporating technology into teaching remains a challenge for many teachers and the degree and type of technology used in the classroom is variable. Reasons for this include teachers' proficiencies in mathematics and their perceptions of the nature of mathematical